Question 37
$A_{1}, A_{2}, \ldots, A_{n}$ are points in the plane. A non-zero real number $k_{i}$ is assigned to each point, so that the square of the distance between $A_{i}$ and $A_{j}$ (for $i \neq j$) is $k_{i} + k_{j}$. Show that $n$ is at most $4$ and that if $n = 4$, then $1 / k_{1} + 1 / k_{2} + 1 / k_{3} + 1 / k_{4} = 0$.